Measurement 43 (2010) 1683–1689
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Frequency dependent parameter model of supercapacitor S. Barsali, M. Ceraolo, M. Marracci, B. Tellini ⇑ Dipartimento di Sistemi Elettrici e Automazione, Università di Pisa, Largo L. Lazzarino 1, I-56122 Pisa, Italy
a r t i c l e
i n f o
Article history: Received 5 August 2009 Received in revised form 24 August 2010 Accepted 3 September 2010 Available online 15 September 2010 Keywords: Experimental analysis Fourier analysis Frequency dependence Permittivity Supercapacitor
a b s t r a c t We present a basic model of supercapacitor based on a theoretical analysis discussing the frequency dependence of the electric permittivity. Directly from the Maxwell’s theory, an equivalent parallel capacitance and conductance is deduced having frequency dependence. Dynamic behavior is assessed in the frequency range up to 20 Hz which is typical for hybrid vehicle applications. In order to investigate the equivalent capacitance and conductance variation in the frequency range 1 mHz–20 Hz, an impedance spectroscopy method has been adopted. We provide a detailed description of the measurement procedure and present the results obtained on a 120 F MaxwellÓ double layer capacitor. Finally, the model response calculated through the Fourier analysis is tested under square and triangular waveforms; results obtained perfectly match the actual measurements. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Supercapacitors are intriguing components for several applications such as power electronics and electric hybrid vehicles [1–3]. Generally speaking, supercapacitors can be quickly charged and discharged, offer high power density and have the advantage of long cycle life [4]. They are an interesting alternative to pulsed batteries for onboard energy storage in hybrid vehicles [5]. Usually, supercapacitors are modeled by RC circuits whose resistance and capacitance are defined on phenomenological basis. An accurate characterization of the charge–voltage relation as well as of the equivalent electrical parameters of the supercapacitor is important [2,3]. This is the basis for a correct understanding and modeling of such devices. As a general consideration, a great effort has been done by the scientific community toward the characterization of dielectric materials. Several time-domain and frequencydomain methods have been developed and applied for ⇑ Corresponding author. E-mail addresses:
[email protected] (S. Barsali), massimo.ceraolo@ ing.unipi.it (M. Ceraolo),
[email protected] (M. Marracci),
[email protected] (B. Tellini). 0263-2241/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2010.09.016
the characterization of permittivity in a wide frequency range. In the microwave frequency range, free-space methods are common measurement techniques [6], while swept-sine and impedance spectroscopy techniques are commonly applied at lower frequencies to reconstruct the dielectric permittivity or the equivalent parameters of the considered capacitor [3,7]. Our aim consisted in performing an experimental analysis to investigate the frequency dispersion of the dielectric medium and to quantify the supercapacitor behavior with a basic model. This, in particular, in a frequency range typical for applications in hybrid vehicles. An inaccurate assessment of the resistive and capacitive components at the various frequencies would lead to a wrong evaluation of the overall system behavior, of its performance and energy efficiency. This paper is organized as follows: in Section 2, we start from the most general definition of permittivity for an isotropic linear dielectric material in the time domain and we recall the concept of complex permittivity for lossy dielectrics. In Section 3, we describe the experimental analysis and the frequency dispersion of the equivalent parallel capacitance C and conductance G. In Section 4, we compare the model behavior with the measurement results under various operating condition.
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C and Cs (F)
120 100 80 60 40 20 0 −3 10
−2
−1
10
10
10
0
1
10
10
2
frequency (Hz) Fig. 2. Variation of C (points) and Cs (triangles) vs. log(f). Frequency 1 mHz–19.5 Hz. Fig. 1. Photograph of the experimental setup adopted for the experimental analysis. The supercapacitor is visible in the bottom-right part of the photo. 120
2. Theoretical analysis The electric permittivity e and the magnetic permeability l are ‘‘macroscopic” parameters: they describe electromagnetic properties of a homogeneous medium over a macroscopic length scale. They relate the macroscopic fields D and E, and B and H that are representative of a spatial averaging of the unobservable microscopic fields over a representative region. If we assume a linear dependence between the electric displacement D and the electric field E, the relevant monochromatic components are related by: 0
00
DðxÞ ¼ eðxÞEðxÞ ¼ e ðxÞ þ ie ðxÞEðxÞ
G and 1/Rs (S)
100 80 60 40 20 0 −3 10
−2
10
10
r ð xÞ _ E ¼ ixðe0 ðxÞ þ ie00 ðxÞÞE_ r H_ ¼ ix e0 ðxÞ i
x
ð2Þ
0
10
1
10
2
10
frequency (Hz)
ð1Þ
where e0 (x) and e00 (x) are even and odd functions, respectively. For sake of more completeness, it is worth to recall that whatever the dielectric material, from physical considerations in the limit of extremely high frequencies, x ? 1, e(x) ? e0. In addition, (1) states that for a dielectric medium with permittivity showing frequency dispersion law the fields D(t) and E(t) are not instantaneously connected, depending D(t) on E(t) through a time convolution relationship [8,9]. From the Ampère-Maxwell’s equation in phasor form for a lossy dielectrics, we can write [9,10]:
−1
Fig. 3. Variation of G (points) and 1/Rs (triangles) vs. log(f). Frequency 1 mHz–19.5 Hz.
Assuming field uniformity inside the dielectric medium the integral form of (2) writes as:
GðxÞ _ I_ ¼ ix CðxÞ i V
ð3Þ
x
and we can adopt an equivalent parallel model of lossy capacitor in order to keep separate the terms of the conduction and displacement current. In particular, the equivalent parallel capacitance C(x) = e0 (x)S/d and conductance
Table 1 Equivalent parallel admittance Y: absolute value and phase vs. f. f (Hz)
jYj (S)
Phase (Y) (deg)
f (Hz)
jYj (S)
Phase (Y) (deg)
0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.3 0.5 0.7 0.9 1.1 1.3
0.7562 1.4122 3.2765 6.3618 12.3738 28.9672 49.0973 80.0417 85.7622 88.6836 90.1113 90.9796 91.5878
82.4012 85.9395 86.5005 84.5882 81.1673 69.3828 54.2286 27.1849 17.2604 14.6511 10.8514 9.6209 9.2377
1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 13.5 16.5 19.5
92.2176 94.4423 95.8047 96.9717 97.7447 98.5391 98.9123 99.3437 99.8785 100.2174 100.9570 101.7296 102.3361
7.6350 6.6859 5.2681 5.0182 4.0548 4.7548 2.7307 3.0817 3.4262 3.7745 4.8196 5.8492 6.8762
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0.5
0.01
5
0.25
0.005
0
−5
charge (C)
10
charge (C)
charge (C)
S. Barsali et al. / Measurement 43 (2010) 1683–1689
0
−0.005
−0.25
−10 −0.1
−0.05
0
0.05
−0.5 −0.02
0.1
0
−0.01
voltage (V)
0
0.01
−0.01 −0.02
0.02
−0.01
voltage (V)
0
0.01
0.02
voltage (V)
Fig. 4. q v relations at frequencies 20 mHz, 50 mHz and 100 mHz (left plot), 0.2 Hz, 0.5 Hz and 1 Hz (center plot), and 2 Hz, 5 Hz and 10 Hz (right plot). Current amplitude was fixed at 1 A.
G(x) = r(x)S/d represent e0 (x) and r(x) apart from the geometrical factor S/d, being S and d the surface and separation gap of the electrodes.
3. Experimental analysis Fig. 1 shows the adopted experimental setup. The source vs is an arbitrary function generator (Agilent 33220A) which is connected to the supercapacitor via a transconductive power amplifier, so that it works as a current generator. A digital storage oscilloscope (Yokogawa DL708E) measures the voltage v across the supercapacitor and the supplied current i via the voltage drop vR = Ri across a R = 0.1 X calibrated resistance. A LabviewÓ platform drives the whole system through the IEEE 488Ó protocol. The device under test is a 120 F MaxwellÓ double layer capacitor. We verified the linear behavior of the supercapacitor in the voltage range 2 + 2.5 V. We then supplied sinusoidal currents having an amplitude of 1 A in the frequency range 1 mHz–19.5 Hz that covers typical frequencies in hybrid vehicle operation and most high power applications [11,12]. From the measured i and v we could estimate the equivalent admittance Y = G + ixC at each frequency and the relevant q,v relation. Absolute value and phase of Y are reported in Table 1.
Figs. 2 and 3 show the C(f) and G(f) vs. log(f). For sake of more completeness and with the aim to provide a comparison between different models, we included in Figs. 2 and 3 the corresponding values of the capacitance Cs and of the inverse 1/Rs of the resistance of an equivalent Cs Rs series model of the supercapacitor. Data relevant to Cs and 1/Rs are obtained from Table 1. These results clearly show a significant frequency dispersion of the permittivity below extremely low frequencies. In particular, from Table 1 it can be easily argued that below 100 mHz the capacitive behavior is dominant, while above 1 Hz the supercapacitor behaves practically as a resistor. It is worth saying that frequency components in the frequency range between 10 mHz and a few hertz are typical in hybrid vehicle operation [11,12]. Thus, an accurate knowledge of the equivalent parameters vs. f becomes very important for a correct modeling and design of such devices. In Fig. 4, we show the dielectric behavior in the q v plane obtained at various frequencies. The amplitude of the supplied currents was always 1 A and frequencies were 20 mHz, 50 mHz and 100 mHz (left plot), 0.2 Hz, 0.5 Hz and 1 Hz (center plot), and 2 Hz, 5 Hz and 10 Hz (right plot). Slopes correspond to the values of C shown in Fig. 2. Although charge and voltage amplitudes are expected to diminish if we increase the frequency at a constant current amplitude, the frequency dispersion of the capacitance C is clearly shown by the rotation of the q v loci.
100
50
90
conductance (S)
capacitance (F)
110 60
40 30 20
80 70 60 50 40
10 0
30 2
4
6
8
10
12
14
16
18
frequency (Hz) Fig. 5. Variation of C vs. f. Frequency range 100 mHz–19.5 Hz.
20
2
4
6
8
10
12
14
16
18
frequency (Hz) Fig. 6. Variation of G vs. f. Frequency range 100 mHz–19.5 Hz.
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in Figs. 5 and 6 in the frequency range 100 mHz–19.5 Hz. As first approximation, above 19.5 Hz we maintained C and G constant. In Figs. 7 and 8, we show the measured (left plots) and calculated (right plots) voltage and current obtained for a triangular current waveform, (peak-to-peak about 2 A) with fundamental frequency 100 mHz and 500 mHz, respectively. Results shown in Fig. 8 confirm a practically resistive behavior at relatively higher operating frequencies. Indeed, the voltage shown in Fig. 8 approximates a triangular waveform as it would be the case by replacing the supercapacitor by an equivalent resistance.
4. Modeling and validation
1
calc. current (A)
meas. current (A)
Usually, equivalent constant and lumped parameter models are used for simulating the supercapacitor behavior with various levels of complexity [13,14]. As discussed in Section 2, we combine a basic parallel model and a Fourier analysis to study the behavior of the supercapacitor under various operating conditions. To check the validity of our model, we choose triangular and square current waveforms having fundamental frequencies of 100 mHz and 500 mHz [16,17]. For all the calculations, we truncated the Fourier series at the 195th harmonic, i.e., 19.5 Hz and 97.5 Hz, respectively. The relevant C and G data are shown
0
−1 0
5
1
0
−1
10
0
5
time (s) 0.02
calc. voltage (V)
meas. voltage (V)
0.02
0
−0.02
10
time (s)
0
−0.02 0
5
10
0
5
10
time (s)
time (s)
1
calc. current (A)
meas. current (A)
Fig. 7. Measured (left plots) and calculated (right plots) results obtained for a triangular current waveform (2 A peak-to-peak – null mean value). Fundamental frequency 100 mHz.
0
−1 0
0.5
1
1.5
1
0
−1
2
0
0.5
time (s)
1.5
2
1.5
2
0.02
calc. voltage (V)
meas. voltage (V)
0.02
0
−0.02
1
time (s)
0
0.5
1
time (s)
1.5
2
0
−0.02
0
0.5
1
time (s)
Fig. 8. Measured (left plots) and calculated (right plots) results obtained for a triangular current waveform (2 A peak-to-peak – null mean value). Fundamental frequency 500 mHz.
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calc. current (A)
meas. current (A)
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10
0
−10 0
5
10
0
−10
10
0
5
time (s) 0.5
calc. voltage (V)
meas. voltage (V)
0.5
0
−0.5
10
time (s)
0
5
0
−0.5
10
0
5
time (s)
10
time (s)
calc. current (A)
meas. current (A)
Fig. 9. Measured (left plots) and calculated (right plots) results obtained for a square current waveform (20 A peak-to-peak – null mean value). Fundamental frequency 100 mHz.
10
0
−10 0
1
10
0
−10 0
2
0.5
1
1.5
2
time (s) 0.2
calc. voltage (V)
meas. voltage (V)
0.2
0
−0.2
0
1
2
0
−0.2
0
1
2
time (s) Fig. 10. Measured (left plots) and calculated (right plots) results obtained for a square current waveform (20 A peak-to-peak – null mean value). Fundamental frequency 500 mHz.
The analogous plots obtained for two square current waveforms (peak-to-peak about 20 A – null mean value) of fundamental frequency 100 mHz and 500 mHz are shown in Figs. 7 and 8. The different current levels play just the role of a scale factor, operating the supercapacitor in linear conditions. Although we can recognize the Gibb’s effect present in the Fourier analysis as a consequence of the adopted finite Fourier series [15], the results confirm the validity of the theoretical and experimental analysis. For sake of more clearness, we show in Fig. 11 the voltage across a pure capacitance through which the same triangular current circulates. Left plot refers to measurement data while right plot to the calculations for which we
adopted C = 8.1 F, i.e., the capacitance value obtained at 500 mHz. Results clearly show the complete inability to model the supercapacitor via an equivalent constant pure capacitance. On the other hand, in Fig. 12, the measured v obtained for triangular and square current waveforms (100 mHz left plot, 500 mHz right plot – solid lines) is compared with that calculated by simulating the capacitor by an equivalent C and G parallel (dashed line). In particular, the capacitance and conductance values have been estimated at the fundamental frequency of the considered current waveforms, i.e., C = 63.4 F and G = 28.7 S for the left plots at 100 mHz, and C = 8.1 F, G = 81.9 S for the right plot at 500 mHz.
S. Barsali et al. / Measurement 43 (2010) 1683–1689
1
calc. current (A)
meas. current (A)
1688
0
−1 0
0.5
1
1.5
1
0
−1
2
0
0.5
1.5
2
1.5
2
0.04
calc. voltage (V)
meas. voltage (V)
0.04
0
−0.04
1
time (s)
time (s)
0
0.5
1
1.5
0
−0.04
2
0
0.5
time (s)
1
time (s)
0.02
calc. voltage (V)
calc. voltage (V)
Fig. 11. Measured results are the same of that shown in Fig. 8. Calculations refer to the case of a pure capacitance C = 8.1 F.
0
−0.02
0
5
0.01
0
−0.01
10
0
time (s) calc. voltage (V)
calc. voltage (V)
0.5
0
−0.5
0
5
1
2
time (s)
10
0.2
0
−0.2
0
1
2
time (s)
time (s)
Fig. 12. Measured and calculated (equivalent C and G parallel) v for triangular (above) and square (below) waveform. Left and right plots are relevant to the 100 mHz and 500 mHz fundamental frequency, respectively. Capacitance and conductance values refer to those estimated at the chosen fundamental frequencies.
It is clear how the representation of the considered supercapacitor by an equivalent constant C and G parallel branch is not adequate to represent its dynamic behavior. In particular, although when triangular waveforms are used the calculated voltage remains close to the measured one, in the case of square waveforms the difference is noticeable. In the latter case the calculated values via the use of equivalent constant C and G reveal a dominant capacitive behavior at high frequencies as there is no discontinuity in the voltage plot. Measurements, instead, highlight a sudden step in the voltage across the capacitor which therefore behaves more as a resistor than as a pure capacitor. On the other hand, this is the result obtained via the use of frequency varying parameters and shown in Figs. 9 and 10. As a final remark, the very good match of the results demonstrates that the choice of a basic model and the
use of the Fourier analysis can represent a valid approach for the study of supercapacitor behavior. 5. Conclusions This paper presents a basic model directly deduced from the Maxwell’s theory for describing the dynamic behavior of supercapacitors. We characterized the frequency response of a supercapacitor up to 20 Hz via an impedance spectroscopy technique. A basic experimental setup was prepared to measure the current through and the voltage across the supercapacitor. In agreement with other works, we observed a significant variation of the equivalent capacitance C and conductance G vs. frequency f between 1 mHz and 19.5 Hz. This suggested us to base calculations on the direct use of the Fourier analysis instead of fitting the observed behavior via more or less
S. Barsali et al. / Measurement 43 (2010) 1683–1689
complex equivalent circuits. The frequency dependent behavior is an important feature to be accounted for while adopting such elements in power electronics and on-board energy storage systems. These applications operate in the considered frequency range and a wrong model would lead to a bad assessment of the overall system performance, mainly in terms of energy efficiency. The obtained results showed a substantially capacitive behavior of the supercapacitor up to about 100 mHz, while above 1 Hz the resistive behavior becomes dominant. Measurements and calculations were in very good agreement, thus confirming the validity of the adopted model and methodology. Acknowledgments This work has been fully supported by Filiera Idrogeno a project funded by Italian Ministry of Education, University and Research, by means of Tuscany Region. References [1] A. Burke, Ultracapacitors: why, how, and where is the technology, J. Power Sources 91 (2000) 37–50. [2] S. Buller, E. Karden, D. Kok, R.W. De Doneker, Modeling the dynamic behavior of supercapacitors using impedance spectroscopy, IEEE Trans. Ind. Appl. 38 (6) (2002) 1622–1626. [3] P.J. Mahon, G.L. Paul, S.M. Keshishian, A.M. Vassallo, Measurement and modeling of the high-power performance of carbon-based supercapacitors, J. Power Sources 91 (2000) 68–76. [4] R. Lu, C. Zhu, L. Tian, Q. Wang, Super-capacitor stacks management system with dynamic equalization technique, IEEE Trans. Magn. 43 (1) (2007) 254–258.
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